5 UPPER AND LOWER INTEGRALS 113 12. Let X be a compact space. Show that in M(X) the set of positive measures with finite support and total mass 1 is dense (with respect to the vague topology) in the set P of all positive measures with total mass 1 . (Let U be a neighborhood of /A e P with respect to the vague topology, consisting of measures yeP such that |/X/i)~~ K/*)l ="» with fi e ^C(X). Consider a continuous partition (gj) of unity and points a} e X such that !/,(*) -E/ifek/WI ^ « for ail /.) 13. Let X be the unit interval [0, 1] in R. Show that the set K of Dirac measures ex (x £ X) is vaguely compact, and that in M(X) the Lebesgue measure lies in the vague closure of the convex hull of K, but does not lie in the convex hull of K. 5. UPPER AND LOWER INTEGRALS WITH RESPECT TO A POSITIVE MEASURE In Sections (13.5) to (13.14) (including the problems), ju denotes a positive measure on a locally compact space X. We shall show (13.7.3) that p. can be extended from «3TR(X) to a vector subspace J$?R(X, p) of Rx, depending on /x and containing (and in general distinct from) JTR(X), in such a way that this extension (also denoted by ju) is a. positive linear form on <£?R(X, fj) (i.e., ^ takes values ^0 on functions ^0 belonging to JSf i(X, //)) and possesses the fundamental property of passage to the limit for increasing sequences: that is to say, if (/„) is an increasing sequence of functions in J$?R(X, //) whose upper envelope/ (12.7.5) also be- longs to j£fR(X, ILL), then we have lim /*(/„) = Let </ (or «/(X)) be the set of all functions /: X-^R which are lower semicontinuous on X and bounded below by a function belonging to JfR(X) (this implies that/(.x) > — oo for all x e X; but we can have/(x) = + oo at some points xeX: indeed, the constant function equal to +00 belongs to ./). Every function/^ 0 which is lower semicontinuous on X belongs to ,/. For every /e ,/, we put (13.5.1) /,*(/)= sup n(g); this is a real number, or +00. Clearly if/e ^TR(X) we have ju*(/) = if/, g E </ and/^ g, we have /**(/) ^ H*(g)\ and for any real number a > 0, we have t**(af) = #ju*(/) for all/e J5". (13.5.2) Let (/„) be an increasing sequence of functions belonging to */, and letf= sup/n (so thatf^J (12.7.6)). Then HUNT PEIJ;SYLVINIA IF(x) = e2nikx.)